Measure estimation on manifolds: an optimal transport approach

Assume that we observe i.i.d. points lying close to some unknown d-dimensional $${\mathcal {C}}^k$$ submanifold M in a possibly high-dimensional space. We study the problem of reconstructing probability distribution generating sample. After remarking this is degenerate for large class standard losses ( $$L_p$$ , Hellinger, total variation, etc.), focus on Wasserstein loss, which build an estimator, based kernel density estimation, whose rate convergence depends d and regularity $$s\le k-1$$ underlying density, but not ambient dimension. In particular, show estimator minimax matches previous rates literature case where manifold cube. The related estimation volume measure loss also considered, exhibited.